Nonlinear Volterra integral equations can arise in systems whenever there is feedback and filtering. Usually these are analyzed by deriving equivalent differential models and then using standard dynamical systems methods. However, the presence of noise in these models usually puts them beyond differential equations, and one must consider the Volterra integral equations directly. The purpose of this paper is to apply a random perturbation method to a canonical network from mathematical neuroscience that has parametric noise, and to demonstrate how the results can be used to estimate the loss of information due to cycle slipping, which corresponds to noise-induced spurious firing in the network.